An Optimal Algorithm to Detect a Line Graph and Output Its Root Graph

@article{Lehot1974AnOA,
  title={An Optimal Algorithm to Detect a Line Graph and Output Its Root Graph},
  author={Philippe G. H. Lehot},
  journal={J. ACM},
  year={1974},
  volume={21},
  pages={569-575}
}
Given a graph <italic>H</italic> with <italic>E</italic> edges and <italic>N</italic> nodes, a graph <italic>G</italic> is sought such that <italic>H</italic> is the line graph of <italic>G</italic>, if <italic>G</italic> exists. The algorithm does this within the order of <italic>E</italic> steps, in fact in <italic>E</italic> + <italic>O</italic>(<italic>N</italic>) steps. This algorithm is optimal in its complexity. 
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References

SHOWING 1-8 OF 8 REFERENCES
The interchange graph of a finite graph
Let G be a finite graph. The interchange graph G' of G, has a vertex corresponding to each edge of G, two vertices of G' being connected if the corresponding edges of G have a common vertex in G. InExpand
Congruent Graphs and the Connectivity of Graphs
We give here conditions that two graphs be congruent and some theorems on the connectivity of graphs, and we conclude with some applications to dual graphs. These last theorems might also be provedExpand
Derived graphs of digraphs
  • Beitrage zur graphentheorie,
  • 1968
Graph theory
Derived graphs of digraphs The interchange graphs of a finite graph
  • Ac~a Math. Acad. Sci. Hungar
  • 1965
The interchange graphs of a finite graph
  • Ac~a Math. Acad. Sci. Hungar
  • 1965